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A train starts from rest and moves with a constant acceleration of 2.0 m/s2 for half a minute. The brakes are then applied and the train comes to rest in one minute. Find
By BYJU'S Exam Prep
Updated on: September 25th, 2023
(a) the total distance moved by train,
(b) the maximum speed attained by the train, and
(c) the position(s) of the train at half the maximum speed.
(a) the total distance moved by train is 2.7 km,
(b) the maximum speed attained by the train is 60 m/s, and
(c) the position(s) of the train at half the maximum speed is 2.25 km.
Step 1: Given that acceleration, a = 2.0 m/s2
Initial velocity, u = 0
Step 2:
Using the equations of motion we have to calculate the final velocity and the total distance traveled.
Final velocity, v = u + at
Distance traveled, x = ut + ½ at2
Step 3: In case of accelerated motion
Given that, a = 2.0 m/s2
Let v1 represent the final speed before applying breaks.
we know that, initial velocity, u1 = 0
Time taken, t1 = ½ x 60 = 30 seconds
Final velocity is now provided by,
v1 = u1 + at1
Substituting the values we get
v1 = 0 + 2 x 30
v1 = 60 m/s
The top speed reached after applying brakes is 60 m/s2
Distance traveled
x = u1t1 + ½ at12
x = 0 + ½ x 2 x 302
x = 900 m
x = 0.9 km
The maximum speed attained is 60 m/s2, Let y be the distance covered while moving at half the maximum speed of 30 m/s.
From the equation now
v12 = u12 + 2ax
302 = 00 + 2 x 2 x y
y = 900/4 = 225 m ….. (1)
As a result, the starting point is 225 metres away.
Step 4: In case of decelerated motion
Final velocity, v2 = 0 m/s
Initial velocity, u2 = 60 m/s
Time taken, t2 = 60 s
From the equation
v2 = u2 + at2
a = (0-60)/60 = -1 m/s2
Distance traveled,
x = u2t2 + ½ at22
Substituting the values we get
x = 60 x 60 + ½ x (-1) x 602
x = 3600 – 1800
x = 1800 m
x = 1.8 km
The total distance covered = 0.9 + 1.8 = 2.7 km
Let z represent the distance travelled in this portion at half the speed.
From
v22 = u22 + 2az
302 = 602 + 2 x (-1) x z
2z = 602 – 302
2z = 2700
z = 1350m
z = 1.35 km …. (2)
Step 5: Conclusion
(a) Total distance traveled by train is given as,
s = x + z
= 900 + 1800
= 2700 m
= 2.7 km
(b) The train can travel at a maximum speed of 60 m/s.
(c) The position of the train is indicated at a speed of half the maximum,
Two places will operate at half the maximum speed (From the point of start),
The position is 225 metres from the starting point.
After braking, the position is = 1.35 km + 0.9 km = 2.25 km
Summary:-
A train starts from rest and moves with a constant acceleration of 2.0 m/s2 for half a minute. The brakes are then applied and the train comes to rest in one minute.
(a) the total distance moved by train is 2.7 km,
(b) the maximum speed attained by the train is 60 m/s, and
(c) the position(s) of the train at half the maximum speed is 2.25 km.
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